Concepts de base
The paper establishes the maximum values of the Sombor-index-like graph invariants SO5 and SO6 within the set of molecular trees with a given number of vertices, and determines the maximum value of SO5 within the set of graphs obtained by applying the join operation to two specific graphs of a given order.
Résumé
The paper focuses on analyzing the maximum values of the Sombor-index-like graph invariants SO5 and SO6 in different classes of graphs.
Key highlights:
The authors establish the maximum values of SO5 and SO6 within the set of molecular trees (trees with maximum degree ≤ 4) with a given number of vertices.
They determine the maximum value of SO5 within the set of graphs obtained by applying the join operation to two specific graphs of a given order.
The proofs rely on connections between number theory, polynomial theory, and multivariable function analysis, covering numerous distinct cases.
Fully classifying the set of connected graphs with the maximum SO5 is identified as a more demanding task, involving the examination of complex multivariable functions and a larger number of cases.
The analysis of univariate functions is reduced to the study of fifth-degree polynomials, which generally do not have solutions in all cases.